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Morse theory for strongly indefinite functional. (English) Zbl 1013.37023

The authors extend some concepts from Morse theory (critical groups, Morse inequalities) to strongly indefinite functionals using a Galerkin approach. They treat functionals \(f\in{\mathcal{C}}^2(H,\mathbb{R})\) defined on a separable Hilbert space \(H\) which have the form \(f(x)=\frac 12\langle Ax,x\rangle+ G(x)\) where \(A\in{\mathcal{L}}(H)\) is bounded linear, selfadjoint and Fredholm, and \(\nabla G\) is compact and Lipschitz continuous on bounded sets. It is also assumed that there exists an approximation scheme \((P_n)_{n\in\mathbb{N}}\) of orthogonal finite-dimensional projectors with \(P_nA-AP_n\rightarrow 0\) in the operator norm. Let \(S\subset H\) be a compact set of critical points which is also an isolated invariant set for the negative gradient flow. In this situation the authors define critical groups \(C_*(f,S)\) and prove Morse inequalities relating \(C_*(f,S)\) and \(C_*(f,S_i)\) if there is a Morse decomposition \(S=\bigcup^m_{i=1} S_i\).

MSC:

37D15 Morse-Smale systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37M99 Approximation methods and numerical treatment of dynamical systems
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